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June 2015 The field $\mathbb F_8$ as a Boolean manifold
René Guitart
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Tbilisi Math. J. 8(1): 31-62 (June 2015). DOI: 10.1515/tmj-2015-0002

Abstract

In a previous paper (“Hexagonal Logic of the Field $\mathbb F_8$ as a Boolean Logic with Three Involutive Modalities”, in The road to Universal Logic), we proved that elements of $\mathbb P(8)$, i.e. functions of all finite arities on the Galois field $\mathbb F_8$, are compositions of logical functions of a given Boolean structure, plus three geometrical cross product operations. Here we prove that $\mathbb P(8)$ admits a purely logical presentation, as a Boolean manifold, generated by a diagram of $4$ Boolean systems of logical operations on $\mathbb F_8$. In order to obtain this result we provide various systems of parameters of the set of unordered bases on $\mathbb F_2^3$, and consequently parametrical polynomial expressions for the corresponding conjunctions, which in fact are enough to characterize these unordered bases (and the corresponding Boolean structures).

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René Guitart. "The field $\mathbb F_8$ as a Boolean manifold." Tbilisi Math. J. 8 (1) 31 - 62, June 2015. https://doi.org/10.1515/tmj-2015-0002

Information

Received: 21 October 2014; Accepted: 21 October 2014; Published: June 2015
First available in Project Euclid: 12 June 2018

zbMATH: 1322.03020
MathSciNet: MR3314180
Digital Object Identifier: 10.1515/tmj-2015-0002

Subjects:
Primary: 03B50
Secondary: 03G05, 06E25, 06E30, 06Exx, 11Txx

Rights: Copyright © 2015 Tbilisi Centre for Mathematical Sciences

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Vol.8 • No. 1 • June 2015
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